Electronic Journal of Qualitative Theory of Differential Equations
2009, No.50, 1-8;http://www.math.u-szeged.hu/ejqtde/
GLOBAL SOLUTIONS FOR ABSTRACT FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS
EDUARDO HERN ´ANDEZ M∗
AND SUELI M. TANAKA AKI
Abstract. In this paper we study the existence of global solutions for a class of abstract functional differential equation with nonlocal conditions. An application is considered.
1. Introduction
In this paper, we discuss the existence of global solutions for an abstract functional differential equation with nonlocal conditions in the form
u′(t) =Au(t) +f(t, ut, u(ρ(t))), t∈I = [0,∞), u0 =g(u, ϕ)∈ B,
(1.1)
whereAis the infinitesimal generator of aC0-semigroup of bounded linear operators
(T(t))t≥0defined on a Banach space (X,k · k); the historyut: (−∞,0]→X, defined
by ut(θ) :=u(t+θ),belongs to some abstract phase space B defined axiomatically, ϕ∈ B and ρ, f, gare appropriated functions.
The study of initial value problems with nonlocal conditions arises to deal spe-cially with some situations in physics. For the importance of nonlocal conditions in different fields, we refer the reader to [1, 2, 3, 4, 5] and the references contained therein.
Differential systems with nonlocal conditions are considered in different works. In Byszewski [1] is studied the existence of mild, strong and classical solutions for the abstract Cauchy problem
x′(t) = Ax(t) + f(t, x(t)), t∈[0, a],
x(0) = x0 + q(t1, t2, t3, ..., tn, x(·))∈X,
where A is the generator of aC0-semigroup of linear operators on a Banach space X;f, q are appropriated functions; 0< t1< .... < tn≤aare prefixed numbers and
the symbol q(t1, t2, t3, ..., tn, x(·)) is used in the sense that the function x(·) can be
evaluated only in the points ti, for instance q(t1, t2, t3, ..., tn, x(·)) =Pni=1αix(ti).
The literature concerning differential equations with nonlocal conditions also in-clude ordinary differential equations, partial differential equations, abstract partial functional differential, integro-differential equations. Related to this matter, we cite
* Corresponding author.
2000 Mathematics Subject Classification. 35R10, 34K30, 49K25, 47D06.
Key words and phrases. Abstract Cauchy problem, semigroup of linear operators.
among others works, [1, 2, 3, 4, 5, 6, 7, 8, 9]. To the best of our knowledge, the study of the existence of “ global ” solutions for equations described in the abstract form (1.1), is a untreated topic, and this fact, is the main motivation of this work.
In this paper,Ais the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (T(t))t≥0 defined on a Banach space (X,k · k) andM, δ
are positive constants such that kT(t)k≤M e−δt, for every t≥0. In this work, the phase space Bis a linear space of functions mapping (−∞,0] into X endowed with a seminorm k · kB and verifying the following axioms.
(A) Ifx: (−∞, a)→X,a >0, is continuous on [0, a) andx0∈ B, then for every
t∈[0, a) the following conditions hold: (i) xt is inB.
(ii) kx(t)k≤H kxtkB.
(iii) kxtkB≤K(t) sup{kx(s)k: 0≤s≤t}+M(t)kx0 kB,
where H >0 is a constant; K, M : [0,∞) →[1,∞), K is continuous, M is locally bounded andH, K, M are independent of x(·).
(A1) For the function x(·) in (A), the function t → xt is continuous from [0, a)
intoB.
(B) The spaceB is complete.
Remark 1. In this paper,Kis a positive constant such that supt≥0{K(t), M(t)} ≤ K. We know from that the functions M(·), K(·) are bounded if, for instance,B is a
fading memory space, see [10, p. 190] for additional details.
Example 1.1. The phase space Cr×Lp(̺,X)
Let r ≥0, 1 ≤ p <∞,and let ̺ : (−∞,−r]7→ R be a nonnegative measurable
function which satisfies the conditions (g-5)-(g-6) of [10]. Briefly, this means that ̺
is locally integrable and there exists a non-negative locally bounded function γ on (−∞,0] such that̺(ξ+θ)≤γ(ξ)̺(θ),for all ξ ≤0 and θ∈(−∞,−r)\Nξ, where Nξ ⊆(−∞,−r) is a set whose Lebesgue measure zero.
The spaceB=Cr×Lp(̺, X) consists of all classes of functions ψ: (−∞,0]7→X
such that ψ is continuous on [−r,0], Lebesgue-measurable, and ̺kψkp is Lebesgue
integrable on (−∞,−r). The seminorm in Cr×Lp(̺, X) is defined as follows:
kψkB:= sup{kψ(θ)k:−r≤θ≤0}+ Z −r
−∞
̺(θ)kψ(θ)kpdθ 1/p
.
The space B= Cr×Lp(̺, X) satisfies axioms (A), (A1), and (B). Moreover, when r = 0 and p = 2, one can then take H = 1, M(t) = γ(−t)1/2 and K(t) =
1 +R0
−t̺(θ)dθ 1/2
for t ≥ 0 (see [10, Theorem 1.3.8]). We also note that if the conditions (g-5)-(g-7) of [10] hold, then B is a uniform fading memory.
For additional details concerning phase space, we cite [10].
In this paper,h: [0,∞)→Ris a positive, continuous and non-decreasing function
such that h(0) = 1 and limt→∞h(t) =∞. In the sequel,C0(X) andCh0(X) are the
spaces defined by
C0(X) = {x∈C([0,∞) :X) : lim
t→∞kx(t)k= 0},
Ch0(X) = {x∈C([0,∞) :X) : lim
t→∞
kx(t)k
h(t) = 0},
endowed with the norms k x k0= supt≥0 kx(t) k and k x kh= supt≥0 kxh((tt)k)
respec-tively. Additionally, we define the space
BCh0(X) ={u∈C(R, X) ;u0 ∈ B, u|[0,∞)∈Ch0(X)}
endowed with the norm kukBC0
h=ku0 kB +ku|[0,∞) kh.
We recall here the following compactness criterion.
Lemma 1.1. A set B ⊂Ch0(X) is relatively compact in Ch0(X) if, and only if,B is equicontinuous, B(t) = {x(t) : x ∈B} is relatively compact in X for all t≥0 and
limt→∞khx((tt)k) = 0 uniformly for x∈B.
2. Existence of mild solutions
In this section, we study the existence of mild solutions for the nonlocal abstract Cauchy problem (1.1). To treat this system, we introduce the following conditions. H1 The functionρ : [0,∞)→[0,∞) is continuous andρ(t)≤tfor every t≥0. H2 The functionf :I× B ×X→X is continuous and there exist an integrable
functionm : [0,∞) → [0,∞) and a nondecreasing continuous function W : [0,∞)→ (0,∞) such that kf(t, ψ, x)k≤m(t)W(kψkB +k xk), for every
(t, ψ, x)∈[0,∞)× B ×X.
H3 The function g : BCh0(X)× B → B is continuous and there exists Lg ≥ 0 such that
kg(u, ϕ)−g(v, ϕ)kB≤Lg ku−vkBC0
h, u, v∈ BC
0
h(X).
H4 The functiong(·, ϕ) :BCh0(X) → B is completely continuous and uniformly bounded. In the sequel,N is a positive constant such thatkg(ψ, ϕ) kB≤N
for everyψ∈ BCh0(X).
In this work, we adopt the following concept of mild solutions for (1.1).
Definition 2.1. A functionu∈C(R, X) is called a mild solution of system (1.1) if u0=g(u, ϕ) and
u(t) =T(t)g(u, ϕ)(0) +
Z t 0
T(t−s)f(s, us, u(ρ(s)))ds, t∈I = [0,∞).
Now, we can establish the principal results of this section.
Theorem 2.1. Assume H1,H2 and H4 be hold and that the followings properties are verified.
(a) The set {T(t)f(s, ψ, x) : (s, ψ, x)∈[0, t]×Br(0,B)×Br(0, X)} is relatively compact in X for everyt≥0 and all r >0.
(b) For everyK >0, limt→∞h1(t)R0tm(s)W(Kh(s))ds= 0.
IfMK(1+H)R∞
0 m(s)ds < R∞
c Wds(s),wherec=KN(1 +H)(1 +M H), then there
exists a mild solution for system (1.1).
Proof. Let Γ :BCh0(I :X)→ BCh0(X) be the operator defined by (Γu)0 =g(u, ϕ)
and
Γu(t) = T(t)g(u, ϕ)(0) +
Z t 0
T(t−s)f(s, us, u(ρ(s)))ds, t≥0.
Next we prove that Γ verifies the conditions in [11, Theorem 6.5.4]. To begin, we note that for u∈ BCh0(X) and t≥0,ku(t)k≤ ku|I khh(t). If Kis the constant in
Remark 1, then
kΓu(t)k
h(t) ≤
M N H h(t) +
M h(t)
Z t 0
m(s)W(2Kkuk
BC0
hh(s))ds,
which from (b) permits us to conclude that Γ is a function from BCh0(X) into BCh0(X).
Let (xn)n∈N be a sequence in BCh0(X) and x ∈ BCh0(X) such that xn → x in
BCh0(X). Let ε >0 and N= supn∈N kxn kBC0
h. From condition (b), we can select
L >0 such that
2M N H h(t) +
M h(t)
Z t 0
m(s)W(2KNh(s))ds≤ ε
2, t≥L. (2.1)
From the properties of the functions f, g, we can chooseNε ∈Nsuch that
kg(xn, ϕ)−g(x, ϕ) kB ≤
ε
4(M H+ 1), (2.2)
Z L 0
kf(s, xns, xn(ρ(s)))−f(s, xs, x(ρ(s)))kds ≤ ε
4M,
(2.3)
for every n≥Nε. Under these conditions, we see that
sup{kΓx
n(t)−Γx(t)k
h(t) :t∈[0, L], n≥Nε} ≤
ε
2. (2.4)
Moreover, for t≥L and n≥Nε, we find that
kΓxn(t)−Γx(t)k
h(t) ≤ M
kg(xn, ϕ)(0)−g(x, ϕ)(0) k
h(t)
+ M
h(t)
Z t 0
kf(s, xns, xn(ρ(s)))−f(s, xs, x(ρ(s)))kds
≤ 2M N H
h(t) +
M h(t)
Z t 0
m(s)W(2KNh(s))ds,
and hence,
sup{kΓx
n(t)−Γx(t)k
h(t) :t≥L, n≥Nε} ≤
ε
2. (2.5)
From the inequalities (2.2), (2.4) and (2.5), it follows that k Γxn−ΓxkBC0 h≤ε,
for every n≥Nε, which proves that Γ is continuous.
To prove that Γ is completely continuous, we introduce the decomposition Γ = Γ1+ Γ2, where (Γ1u)0=g(u, ϕ), (Γ2u)0 = 0 and
Γ1u(t) = T(t)g(u, ϕ)(0), t≥0,
(2.6)
Γ2u(t) = Z t
0
T(t−s)f(s, us, u(ρ(s)))ds, t≥0.
(2.7)
From Lemma 1.1 and the properties of the semigroup (T(t))t≥0, it is easy to see that
Γ1 is completely continuous. Next, we prove that Γ2 is also completely continuous.
From [12, Lemma 3.1], we infer that Γ2Br(0,BCh0)|[0,a]={Γ2x|[0,a];x∈Br(0,BC
0
h)}
is relatively compact in C([0, a], X) for every a > 0. Considering this property, Lemma 1.1 and the fact that
kΓ2x(t)k h(t) ≤
M h(t)
Z t 0
m(s)W(2Krh(s))ds→0, as t→ ∞,
uniformly for x∈Br(0,BCh0), we can conclude that the set Γ2(Br(0,BCh0)) is
rela-tively compact in BCh0(X) for everyr >0. Thus, Γ is completely continuous. To finish the proof, we obtain a priori estimates for the solutions of the integral equation u = λΓu, λ ∈ (0,1)}. Let λ ∈ (0,1) and uλ ∈ BCh0(X) be a solution of
λΓz=z. By using the notation αλ(t) =K(sups∈[0,t]kuλ(s)k +kuλ0 kB), for t∈I
we see that
kuλ(t)k ≤ M N H+M Z t
0
m(s)W(kuλs kB +kuλ(ρ(s))k)ds,
≤ M N H+M Z t
0
m(s)W((1 +H)αλ(s))ds,
and hence,
αλ(t) ≤ Kkg(u, ϕ)kB +KM N H+KM Z t
0
m(s)W(1 +H)αλ(s))ds,
≤ KN(1 +M H) +KM Z t
0
m(s)W((1 +H)αλ(s))ds.
Defining by βλ the right hand side of the last inequality, we see that βλ′(t) ≤ KM m(t)W((1 +H)βλ(t)),which implies that
Z (1+H)βλ(t)
(1+H)βλ(0)=c
ds
W(s) ≤MK(1 +H)
Z ∞ 0
m(s)ds < Z ∞
c ds W(s).
This inequality permits us to conclude that the set of functions {βλ :λ∈(0,1)}
is bounded in C(R) and as consequence that the set{uλ :λ∈(0,1)} is bounded in BCh0(X).
Now the assertion is a consequence of [11, Theorem 6.5.4]. The proof is complete.
Theorem 2.2. Assume assumptions H1, H2 andH3 be hold. If the conditions (a)
and (b) of Theorem 2.1 are valid and
(2.8) Lg(1 +M H) +Mlim inf
r→∞ Z +∞
0
m(s)W((1 + 2K)rh(s))
rh(s) ds <1,
then there exists a mild solution of (1.1).
Proof. Let Γ,Γ1,Γ2 be the operators introduced in the proof of Theorem 2.1. We
claim that there exists r >0 such that Γ(Br)⊂Br, where Br=Br(0,BCh0(X)). In
fact, if this property is false, then for every r > 0, there exist xr ∈ Br and tr ≥0 such that r <k(Γxr)0kB +k Γx
r(tr)
h(tr) k. Consequently,
r ≤ kg(xr, ϕ)kB +M
kg(xr, ϕ)(0)k
h(tr)
+ M
h(tr) Z t
0
m(s)W(Kkxr
0kB +K sup
θ∈[0,s]
kxr(θ)k+kxr(ρ(s))k)ds
≤ Lg kxrkBC0
h +kg(0, ϕ) kB +
M H h(tr)Lg kx
rk
BC0 h
+ M
h(tr) kg(0, ϕ)kB + M h(tr)
Z tr 0
m(s)W(Kr+Krh(s) +rh(s))ds
≤ Lg(1 +M H)r+ (1 +M)kg(0, ϕ)kB+M Z tr
0
m(s)((2K+ 1)rh(s)) h(s) ds
and then,
1≤Lg(1 +M H) +Mlim inf r→∞
Z +∞ 0
m(s)W((2K+ 1)rh(s)) rh(s) ds,
which is contrary to (2.8).
Let r >0 such that Γ(Br)⊂Br. From the proof of Theorem 2.1, we know that
Γ2 is a completely continuous on Br. Moreover, from the estimate,
kΓ1u−Γ1vkBC0 h
≤ Lg ku−vkBC0
h +Hkg(u, ϕ)−g(v, ϕ) kB≤Lg(1 +H)ku−vkBCh0,
we infer that Γ1 is a contraction onBrfrom which we conclude that Γ is a condensing
operator on Br.
The assertion is now a consequence of [13, Theorem 4.3.2].
3. An application
To complete this work, we study the existence of global solutions for a concrete partial differential equation with nonlocal conditions. In the sequel, X = L2[0, π] and A:D(A) ⊂X → X is the operator Ax=x′′ with domain D(A) := {x ∈ X :
x′′∈X, x(0) =x(π) = 0}.It is well known thatA is the infinitesimal generator of an analytic semigroup (T(t))t≥0 on X. Furthermore, A has discrete spectrum with
eigenvalues of the form −n2, n ∈ N, and corresponding normalized eigenfunctions
given by zn(x) = (π2) 1
2sin(nx); the set of functions{zn :n∈ N} is an orthonormal
basis for X and T(t)x=P∞ n=1e−n
2t
hx, znizn for every x ∈ X and all t ≥ 0. It
follows from this last expression that (T(t))t≥0 is a compact semigroup on X and
that kT(t)k≤e−t for every t≥0.
As phase space, we choose the space B=C0×L2(̺, X), see Example 1.1, and we
assume that the conditions (g5) and (g7) in [10] are verified. Under these conditions, B is a fading memory space, and as consequence, there exists K > 0 such that supt≥0{M(t), K(t)} ≤K.
Consider the delayed partial differential equation with nonlocal conditions
∂
∂tw(t, ξ) = ∂2
∂ξ2w(t, ξ) + Z t
−∞
a(t, s−t)w(s, ξ)ds+b(t, w(ρ(t), ξ)),
(3.1)
w(t,0) = w(t, π) = 0, t≥0,
(3.2)
w(s, ξ) =
∞ X i=1
Liw(ti+s, ξ) +ϕ(s, ξ), s≤0, ξ ∈[0, π],
(3.3)
fort≥0 andξ ∈[0, π], and where (Li)i∈N,(ti)i∈Nare sequences of real numbers and ϕ∈ B.
To treat this system in the abstract form (1.1), we assumeP∞
i=1 |Li |h(ti)<∞,
the functions ρ: [0,∞) →[0,∞), a:R2 → Rare continuous, ρ(t) ≤t and L1(t) =
(R0 −∞
a2(t,s)
̺(s) ds)1/2 < ∞,for every t≥ 0. We also assume that b : [0,∞)×R → R
is continuous and that there exists a continuous function L2 : [0,∞)→ [0,∞) such
that |b(t, x)| ≤L2(t)|x|, for everyt≥0 and x∈R.
By defining the functions g:BCh0(X)→ B,f : [0,∞)× B ×X→X by g(u, ϕ) =
ϕ+P∞
i=1Liuti and
f(t, ψ, x)(ξ) =
Z 0 −∞
a(t, s)ψ(s, ξ)ds+b(t, x(ρ(t), ξ)),
we can transform system (3.1)-(3.3) into the abstract system (1.1). Moreover, it is easy to see that f, g are continuous, k f(t, ψ, x) k≤L1(t) k ψ kB +L2(t) k x k, for
all (t, ψ, x)∈[0,∞) and g verifies conditions H3 withLg =KP∞i=1Lih(ti).
The next result is a direct consequence of Theorem 2.2. We omit the proof.
Proposition 3.1. IfKP∞
i=1Lih(ti) + (1 + 2K) R∞
0 (L1(s) +L2(s))ds <1,then there
exists a mild solution of (3.1)-(3.3).
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(Received May 5, 2009)
Eduardo Hern´andez M &Sueli M. Tanaka Aki
Departamento de Matem´atica, Instituto de Ciˆencias Matem´aticas de S˜ao Carlos, Uni-versidade de S˜ao Paulo, Caixa Postal 668, 13560-970 S˜ao Carlos, SP, Brazil
E-mail address: lalohm@icmc.usp.br
E-mail address: smtanaka@icmc.usp.br
